Argumentation has proven successful in a number of domains, including multi-agent systems and decision support in medicine and engineering. We propose its application to a domain yet largely unexplored by argumentation research: computational linguistics. Over the past decade or so advances in this field have commonly relied on data-driven solutions, i.e. machine learning. Recently, however, there appears to be a growing consent that, in order to achieve significant advances in certain areas of artificial intelligence, in general, and computational linguistics, in particular, we may need to consider data-driven approaches in unison with reasoning-, logic- and rule-based solutions. To this end we have developed a novel classification methodology that incorporates reasoning through argumentation with supervised classifiers. We train classifiers and then argue about the validity of their output. To do so we identify arguments that formalise prototypical knowledge of a problem and use them to correct misclassifications. We thus have at our disposal multiple ways of incorporating knowledge in classification and are able to integrate knowledge as it becomes available, without the need to retrain classifiers. We illustrate our methodology on two tasks. On the one hand we address binary cross-domain sentiment polarity classification, where we train classifiers on one corpus, e.g. Tweets, to identify positive/negative polarity, and classify instances from another corpus, e.g. sentences from movie reviews. On the other hand we address a form of argumentation mining that we call Relation-based Argumentation Mining, where we classify pairs of sentences based on whether the first sentence attacks or supports the second, or whether it does neither. Whenever we find that one sentence attacks/supports the other we consider both to be argumentative, irrespective of their stand-alone argumentativeness. For both tasks we improve classification performance when using our methodology, compared to using standard classifiers, only.